Do regular Wiimotes work on the Wii-U

Now the answer for this question is Yes (with additional conditions though). Latest system update (September 30, 2013) allows all Wii games to be played on Wii U GamePad controller, however original Wii Remote controller is required for games control. Reference.

I would normally refer to the printed manual, which we do of course have access to in English, but it refers to the electronic manual for more information about USB devices, and this manual only seems to exist on the Wii U itself.

I don't have a Wii U yet (partly because it isn't out here yet), but I would expect that this information is also repeated there, in which case an American Wii U owner should be able to verify this.

At any rate, I watched the video, and based on my limited knowledge of Japanese, this seems to relate to the notice shown at 8:16:

大きな電流が流れてしまう可能性があり
「Yケーブルタイプ」でないと
Wii Uでの安定動作が保証できない

If my translation is reasonably accurate, they're essentially saying that drives requiring a Y-cable connection (an extra USB plug for power) may not function correctly. 2.5" drives which don't require this should be fine (note that some drives essentially negotiate for more power over a single port; I suspect that the Wii U does not support this, so those drives would neeed a Y-cable to get 1A).

Most likely, they're just being cautious, and your drive will work fine, but it is possible that there is some edge case where the Wii U is unable to power both ports and ends up disabling the "extra" port.

However, do note that this is entirely based on the text in the video - I don't understand spoken Japanese, so Iwata may be saying something additional here which affects the meaning; and there is the possiblity that I mistranslated something.